$\"\"$

Consider the two numbers be $\"\"$ and $\"\"$ .

The sum of two positive numbers is $\"\"$ .

$\"\"$

Consider the function be the sum of the squares of two numbers is $\"\"$ .

$\"\"$

The smallest number is possible when first derivative function becomes zero.

Apply derivative on each side with respect to $\"\"$ .

$\"\"$

Equate $\"\"$ to zero.

$\"\"$

Substitute $\"\"$ in $\"\"$ .

$\"\"$

Therefore the smallest value of sum of the squares of two numbers is $\"\"$ .

$\"\"$

The smallest value of sum of the squares of two numbers is $\"\"$ .